In this paper, we determine the spectral instability of periodic odd waves for the defocusing fractional cubic nonlinear Schrödinger equation. Our approach is based on periodic perturbations that have the same period as the standing wave solution, and we construct real periodic waves by minimizing a suitable constrained problem. The odd solution generates three negative simple eigenvalues for the associated linearized operator, and we obtain all this spectral information by using tools related to the oscillation theorem for fractional Hill operators. Newton’s iteration method is presented to generate the odd periodic standing wave solutions and numerical results have been used to apply the spectral stability theory via Krein signature as established in Kapitula et al. (2004) and Kapitula et al. (2005).

中文翻译：

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散焦分数三次非线性薛定谔方程的周期波稳定性

在本文中，我们确定了散焦分数三次非线性薛定谔方程的周期性奇波的谱不稳定性。我们的方法基于与驻波解具有相同周期的周期性扰动，并且我们通过最小化合适的约束问题来构造真实的周期波。奇数解为相关线性化算子生成三个负简单特征值，并且我们通过使用与分数希尔算子振荡定理相关的工具获得所有这些谱信息。提出了牛顿迭代方法来生成奇周期驻波解，并且数值结果已用于通过 Kapitula 等人建立的 Kerin 签名应用光谱稳定性理论。 (2004) 和卡皮图拉等人。 （2005）。